A triangulated category T whose suspension functor Σ satisfies Σm≃IdT as additive functors is called an
Nearby in the stack
m
-periodic triangulated category. Such a category does not have a tilting object by the periodicity. In this paper, we introduce the notion of an
m
-periodic tilting object in an
m
-periodic triangulated category, which is a periodic analogue of a tilting object in a triangulated category, and prove that an
m
-periodic triangulated category having an
m
-periodic tilting object is triangulated equivalent to the
m
-periodic derived category of an algebra under some homological assumptions. As an application, we construct a triangulated equivalence between the stable category of a self-injective algebra and the
m
-periodic derived category of a hereditary algebra.